<!DOCTYPE html>
<html lang="zh-CN">
  <head>
    <meta charset="utf-8">
    <meta name="viewport" content="width=device-width,initial-scale=1">
    <title>Material of Analysis | 概率和概率分布</title>
    <meta name="description" content="The analytical data that you often use to learn">
    <link rel="icon" href="/material/logo.png">
  <link rel="manifest" href="/material/manifest.json">
  <meta name="theme-color" content="#3eaf7c">
  <meta name="apple-mobile-web-app-capable" content="yes">
  <meta name="apple-mobile-web-app-status-bar-style" content="black">
  <link rel="apple-touch-icon" href="/material/icons/apple-touch-icon-152x152.png">
  <link rel="mask-icon" href="/material/icons/safari-pinned-tab.svg" color="#3eaf7c">
  <meta name="msapplication-TileImage" content="/icons/msapplication-icon-144x144.png">
  <meta name="msapplication-TileColor" content="#000000">
    
    <link rel="preload" href="/material/assets/css/0.styles.89749010.css" as="style"><link rel="preload" href="/material/assets/js/app.671f232e.js" as="script"><link rel="preload" href="/material/assets/js/13.17c8fcb3.js" as="script"><link rel="prefetch" href="/material/assets/js/9.6ad28882.js"><link rel="prefetch" href="/material/assets/js/1.a893eea4.js"><link rel="prefetch" href="/material/assets/js/2.ad32dd32.js"><link rel="prefetch" href="/material/assets/js/3.2af42d2c.js"><link rel="prefetch" href="/material/assets/js/4.b8a50edf.js"><link rel="prefetch" href="/material/assets/js/5.3a34f633.js"><link rel="prefetch" href="/material/assets/js/6.36baf8b1.js"><link rel="prefetch" href="/material/assets/js/7.23741c7b.js"><link rel="prefetch" href="/material/assets/js/8.0b506f6a.js"><link rel="prefetch" href="/material/assets/js/10.aa5bc1f8.js"><link rel="prefetch" href="/material/assets/js/11.b5d01b0b.js"><link rel="prefetch" href="/material/assets/js/12.2aab57fe.js"><link rel="prefetch" href="/material/assets/js/14.b0d8de38.js"><link rel="prefetch" href="/material/assets/js/15.95c4b4d8.js"><link rel="prefetch" href="/material/assets/js/16.8279b098.js"><link rel="prefetch" href="/material/assets/js/17.c88c3a9c.js"><link rel="prefetch" href="/material/assets/js/18.0635186f.js"><link rel="prefetch" href="/material/assets/js/19.5e722a99.js">
    <link rel="stylesheet" href="/material/assets/css/0.styles.89749010.css">
  </head>
  <body>
    <div id="app" data-server-rendered="true"><div class="theme-container"><header class="navbar"><div class="sidebar-button"><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" role="img" viewBox="0 0 448 512" class="icon"><path fill="currentColor" d="M436 124H12c-6.627 0-12-5.373-12-12V80c0-6.627 5.373-12 12-12h424c6.627 0 12 5.373 12 12v32c0 6.627-5.373 12-12 12zm0 160H12c-6.627 0-12-5.373-12-12v-32c0-6.627 5.373-12 12-12h424c6.627 0 12 5.373 12 12v32c0 6.627-5.373 12-12 12zm0 160H12c-6.627 0-12-5.373-12-12v-32c0-6.627 5.373-12 12-12h424c6.627 0 12 5.373 12 12v32c0 6.627-5.373 12-12 12z"></path></svg></div><a href="/material/" class="home-link router-link-active"><!----><span class="site-name">
      Material of Analysis
    </span></a><div class="links"><div class="search-box"><input aria-label="Search" autocomplete="off" spellcheck="false" value=""><!----></div><nav class="nav-links can-hide"><div class="nav-item"><a href="/material/basic/" class="nav-link router-link-active">基础</a></div><a href="https://github.com/docschina/vuepress" target="_blank" rel="noopener noreferrer" class="repo-link">
    GitHub
    <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path><polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></nav></div></header><div class="sidebar-mask"></div><div class="sidebar"><nav class="nav-links"><div class="nav-item"><a href="/material/basic/" class="nav-link router-link-active">基础</a></div><a href="https://github.com/docschina/vuepress" target="_blank" rel="noopener noreferrer" class="repo-link">
    GitHub
    <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path><polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></nav><ul class="sidebar-links"><li><div class="sidebar-group first"><p class="sidebar-heading open"><span>基础</span><!----></p><ul class="sidebar-group-items"><li><a href="/material/basic/" class="sidebar-link">介绍</a></li><li><a href="/material/basic/getting-started.html" class="sidebar-link">数据的描述</a></li><li><a href="/material/basic/basic-config.html" class="active sidebar-link">概率和概率分布</a><ul class="sidebar-sub-headers"><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#如何应用概率进行推断" class="sidebar-link">如何应用概率进行推断</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#确定一个事件的概率" class="sidebar-link">确定一个事件的概率</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#基本的事件关系和概率法则" class="sidebar-link">基本的事件关系和概率法则</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#条件概率和独立性" class="sidebar-link">条件概率和独立性</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#bayes公式" class="sidebar-link">Bayes公式</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#离散变量和连续变量" class="sidebar-link">离散变量和连续变量</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#一个常见得离散随机变量：二项分布" class="sidebar-link">一个常见得离散随机变量：二项分布</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#连续随机变量得概率分布" class="sidebar-link">连续随机变量得概率分布</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#一个常见得连续随机变量-正态分布" class="sidebar-link">一个常见得连续随机变量:正态分布</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#随机抽样" class="sidebar-link">随机抽样</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#抽样分布" class="sidebar-link">抽样分布</a></li><li class="sidebar-sub-header"><a href="/material/basic/basic-config.html#二项分布得正态逼近" class="sidebar-link">二项分布得正态逼近</a></li></ul></li><li><a href="/material/basic/assets.html" class="sidebar-link">关于总体中心值的推断</a></li></ul></div></li></ul></div><div class="page"><div class="content"><h1 id="概率和概率分布"><a href="#概率和概率分布" aria-hidden="true" class="header-anchor">#</a> 概率和概率分布</h1><h2 id="如何应用概率进行推断"><a href="#如何应用概率进行推断" aria-hidden="true" class="header-anchor">#</a> 如何应用概率进行推断</h2><p>概率第一种解释叫做<strong>概率得古典解释</strong>,其中，每种可能的不同结果称为一个<strong>基本事件</strong></p><p>一个<strong>事件</strong>被认为是许多基本事件的集合</p><p>在概率的古典解释下，一个事件<math><mi>E</mi></math>的概率是用有利于事件E的基本事件数<math><msub><mi>N</mi><mi>e</mi></msub></math>，与所有可能的基本事件总数N的比值来计算的:</p><math display="block"><mi>P</mi><mi>（</mi><mi>事</mi><mi>件</mi><mi>E</mi><mi>）</mi><mo>=</mo><mfrac><msub><mi>N</mi><mi>e</mi></msub><mi>N</mi></mfrac></math><p>这种解释的适用性取决于所有基本事件都是等可能得假设</p><p>概率的第二种解释被称为概率的频率概念，是定义概率的一种经验方法</p><p>用符号表示,如果试验进行了<math><mi>n</mi></math>次，并且在这些试验中事件<math><mi>E</mi></math>发生了<math><msub><mi>n</mi><mi>e</mi></msub></math>次，则事件<math><mi>E</mi></math>的概率近似于</p><math display="block"><mi>P</mi><mi>（</mi><mi>事</mi><mi>件</mi><mi>E</mi><mi>）</mi><mo>≈</mo><mfrac><msub><mi>n</mi><mi>e</mi></msub><mi>n</mi></mfrac></math><p>我们子所以说是“近似于”，是因为我们认为，确切的频率<math><mi>P</mi><mi>（</mi><mi>事</mi><mi>件</mi><mi>E</mi><mi>）</mi></math>是在对现象进行了大量的观察或者重复时事件E发生的频率</p><p>第三种是根据<strong>个人或者主观的概念</strong></p><h2 id="确定一个事件的概率"><a href="#确定一个事件的概率" aria-hidden="true" class="header-anchor">#</a> 确定一个事件的概率</h2><p>我们将用概率的古典解释和频率的概念说明基本事件或事件的概率的计算方法</p><h2 id="基本的事件关系和概率法则"><a href="#基本的事件关系和概率法则" aria-hidden="true" class="header-anchor">#</a> 基本的事件关系和概率法则</h2><p>一个事件，比如说事件A得概率总满足性质</p><math display="block"><mn>0</mn><mo>≤</mo><mi>P</mi><mi>（</mi><mi>A</mi><mi>）</mi><mo>≤</mo><mn>1</mn></math><p>即一个事件的概率在0(该事件的发生是不可能得)到1(该事件的发生是必然的)的范围内</p><p><strong>互相排斥</strong>: 如果一个事件A的发生排除了另一事件B发生的可能，称事件A和B是互相排斥的</p><p>如果两个事件A和B互相排斥，则事件&quot;A或B发生&quot;的<strong>概率</strong>是</p><math display="block"><mi>P</mi><mi>（</mi><mi>A</mi><mi>或</mi><mi>B</mi><mi>）</mi><mo>=</mo><mi>P</mi><mi>（</mi><mi>A</mi><mi>）</mi><mo>+</mo><mi>P</mi><mi>（</mi><mi>B</mi><mi>）</mi></math><p>互相排斥事件概率的可加性的定义可推广到两个以上的事件</p><p><strong>事件A</strong>: 事件A的补定义为&quot;事件A不发生&quot;这样的一个事件，A的补用符号<math><mover><mi>A</mi><mo accent="true">‾</mo></mover></math>表示，</p><p>这样，如果我们定义事件A的补作为一个新的事件，也就是&quot;A不发生&quot;,则得出下列结论</p><math display="block"><mi>P</mi><mi>（</mi><mi>A</mi><mi>）</mi><mo>+</mo><mi>P</mi><mi>（</mi><mover><mi>A</mi><mo accent="true">‾</mo></mover><mi>）</mi><mo>=</mo><mn>1</mn></math><p><strong>概率的性质</strong>
如果一个试验中，A和B是两个互相排斥的事件，那么P(A)和P(B)必须满足下列性质</p><ol><li><math><mn>0</mn><mo>≤</mo><mi>P</mi><mi>（</mi><mi>A</mi><mi>）</mi><mo>≤</mo><mn>1</mn><mi>和</mi><mn>0</mn><mo>≤</mo><mi>P</mi><mi>（</mi><mi>B</mi><mi>）</mi><mo>≤</mo><mn>1</mn></math></li><li><math><mi>P</mi><mi>（</mi><mi>A</mi><mi>或</mi><mi>B</mi><mi>发</mi><mi>生</mi><mi>）</mi><mo>=</mo><mi>P</mi><mi>（</mi><mi>A</mi><mi>）</mi><mo>+</mo><mi>P</mi><mi>（</mi><mi>B</mi><mi>）</mi></math></li><li><math><mi>P</mi><mi>（</mi><mi>A</mi><mi>）</mi><mo>+</mo><mi>P</mi><mi>（</mi><mover><mi>A</mi><mo accent="true">‾</mo></mover><mi>）</mi><mo>=</mo><mn>1</mn><mi>和</mi><mi>P</mi><mi>（</mi><mi>B</mi><mi>）</mi><mo>+</mo><mi>P</mi><mi>（</mi><mi>B</mi><mi>）</mi><mo>=</mo><mn>1</mn></math></li></ol><p>现在我们定义另外两个事件之间的关系:<strong>两个事件并和交</strong></p><h2 id="条件概率和独立性"><a href="#条件概率和独立性" aria-hidden="true" class="header-anchor">#</a> 条件概率和独立性</h2><h2 id="bayes公式"><a href="#bayes公式" aria-hidden="true" class="header-anchor">#</a> Bayes公式</h2><h2 id="离散变量和连续变量"><a href="#离散变量和连续变量" aria-hidden="true" class="header-anchor">#</a> 离散变量和连续变量</h2><h2 id="一个常见得离散随机变量：二项分布"><a href="#一个常见得离散随机变量：二项分布" aria-hidden="true" class="header-anchor">#</a> 一个常见得离散随机变量：二项分布</h2><h2 id="连续随机变量得概率分布"><a href="#连续随机变量得概率分布" aria-hidden="true" class="header-anchor">#</a> 连续随机变量得概率分布</h2><h2 id="一个常见得连续随机变量-正态分布"><a href="#一个常见得连续随机变量-正态分布" aria-hidden="true" class="header-anchor">#</a> 一个常见得连续随机变量:正态分布</h2><h2 id="随机抽样"><a href="#随机抽样" aria-hidden="true" class="header-anchor">#</a> 随机抽样</h2><h2 id="抽样分布"><a href="#抽样分布" aria-hidden="true" class="header-anchor">#</a> 抽样分布</h2><h2 id="二项分布得正态逼近"><a href="#二项分布得正态逼近" aria-hidden="true" class="header-anchor">#</a> 二项分布得正态逼近</h2><hr><blockquote><p>原文：<a href="https://vuepress.vuejs.org/guide/basic-config.html" target="_blank" rel="noopener noreferrer">https://vuepress.vuejs.org/guide/basic-config.html</a></p></blockquote><hr></div><div class="content edit-link"><a href="https://github.com/docschina/vuepress/edit/master/docs/basic/basic-config.md" target="_blank" rel="noopener noreferrer">在GitHub上编辑此页</a><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path><polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></div><div class="content page-nav"><p class="inner"><span class="prev">
        ← <a href="/material/basic/getting-started.html" class="prev">
          数据的描述
        </a></span><span class="next"><a href="/material/basic/assets.html">
          关于总体中心值的推断
        </a> →
      </span></p></div></div></div></div>
    <script src="/material/assets/js/13.17c8fcb3.js" defer></script><script src="/material/assets/js/app.671f232e.js" defer></script>
  </body>
</html>
